MEN OF MATHEMATICS
In sharp distinction to this is the other phrase which is reiter-
ated over and over again in mathematical writing: 'There
exists.' For example, some would say 'there exists a whole
number greater than 2', or 'there exists a number less than n
and greater than n — 2.' The use of such phraseology definitely
commits its user to the creed which Kronecker held to be
untenable, unless, of course, the 'existence1 is proved by a con-
struction. The existence is not proved for the sets (as defined
above) which appear in Cantor's theory.
These two ways of speaking divide mathematicians into two
types: the *we can' men believe (possibly subconsciously) that
mathematics is a purely human invention; the * there exists'
men believe that mathematics has an extra-human 'existence'
of its own, and that *we' merely come upon the 'eternal truths'
of mathematics in our journey through life, in much the same
way that a man taking a walk in a city comes across a number
of streets with whose planning he had nothing whatever to do.
Theologians are 'exist' men; cautious sceptics for the most
part 'we' men. 'There exist an infinity of even numbers, or of
primes', say the advocates of extra-human 'existence'; 'produce
them', say Kronecker and the 4we* men.
That the distinction is not trivial can be seen from a famous
instance of it in the New Testament. Christ asserted that the
Father 'exists'; Philip demanded 'Show us the Father and it
sufficeth us.' Cantor's theory is almost wholly on the 'existence'
side. Is it possible that Cantor's passion for theology deter-
mined his allegiance? If so, we shall have to explain why
Kronecker, also a connoisseur of Christian theology, was the
rabid *we' man that he was. As in all such questions ammunition
for either side can be filched from any pocket.
A striking and important instance of the 'existence' way of
looking at the theory of sets is afforded by what is known as
Zermelo's postulate (stated in 1904). 'For every set M whose
elements are sets P (that is, M is a set of sets, or a class of
classes), the sets P being non-empty and non-overlapping (no
two contain things in common), there exists at least one set fi
which contains precisely one element from each of the sets P
which constitute M.' Comparison of this with the previously
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